---
tag: QSTN8NMSK0
kind: definition
name: Join
msc: [54B17, 55P99]
defines: ZEETZHV69Y
uses: [0NQ1H2X495, 0GBG8B49AA]
sources:
  - url: https://en.wikipedia.org/wiki/Join_(topology)
---

## Definition

Let $X$ and $Y$ be topological spaces and let $I = [0, 1]$.
The *join* of $X$ and $Y$ is the quotient space
$$X * Y = (X \times Y \times I) / \sim,$$
where $X \times Y \times I$ has the [[product topology|0GBG8B49AA]], and $X * Y$ carries the [[quotient topology|0NQ1H2X495]].
The relation $\sim$ is the equivalence relation generated by two rules.
First, $(x, y, 0) \sim (x, y', 0)$ for all $x \in X$ and $y, y' \in Y$.
Second, $(x, y, 1) \sim (x', y, 1)$ for all $x, x' \in X$ and $y \in Y$.

## Remarks

The class of $(x, y, t)$ is often written $(1 - t) x + t y$, since it lies on the segment from $x$ to $y$.
The images of $t = 0$ and $t = 1$ are copies of $X$ and of $Y$ in $X * Y$.
Example: if $Y$ is one point, then $X * Y$ is the cone on $X$.
Example: if $Y = S^0$ is two points, then $X * S^0$ is the unreduced [[suspension|Q7RFNJH76Q]] of $X$.
Example: $S^0 * S^0$ is homeomorphic to the circle $S^1$, a square with four vertices.
Non-example: the product $X \times Y \times I$ with no identification is not the join; for $X = Y = S^0$ it is four disjoint segments, while $S^0 * S^0$ is a circle.